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Full text of "Analytical Mechanics"

MOTION OF A PARTICLE                       131
Let   v = VQ when t = 0, then ec = VQ  2.    Therefore
k
k
or
(3)
LIMITING VELOCITY.  The last equation has a simple interpretation which comes out clearly by plotting the time as abscissa and the velocity as ordinate. There are four special cases which depend upon the following values of the initial velocity :
(a) VQ = 0,    (b) VQ <   >
(c) t;0=,   (d)
.
Curves (a), (b), (c), and (d) of Fig. 69 represent these cases.
It is evident from these curves that whatever its initial value the velocity tends to the same
limiting value ~, called the lim-
kr
iting velocity. In the third case the velocity remains constant, as shown by the horizontal line (c), because the resisting force exactly balances the moving force. Integrating equation (3) we get
o
FIG. 69.
Let s = 0 when t = 0, then
Therefore
(4)